



 
Hi Sandra, In fact if k < 0 then x^{3}  3kx + 1 = 0 only has one root. x^{3}  3kx + 1 = 0 is the same as x(x^{2}  3k) = 1 Now if k is negative, then x^{2}  3k must be positive. So x times some positive number equals 1. Thus x is negative. Thus if k < 0 then every root of x^{3}  3kx + 1 = 0 is negative. Suppose there are two roots a and b, then (x  a) and (x  b) are factors of x^{3}  3kx + 1 and hence
for some polynomial g(x). Since the original polynomial is a cubic, g(x) is linear and since the leading coefficient of the cubic is 1, g(x) = x  c for some real number c. Hence
Thus a, b and c are roots of the cubic so they are all negative. What do you know about a b c? Stephen and Penny
 


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